If tan$\theta$ and sec$\theta$ are the roots of the equation $a{x}^2+bx+c=0$ then 

$\begin{array}{l}\tan \theta +\sec \theta =-\frac{b}{a},\tan \theta \sec \theta =\frac{c}{a}\\ \therefore \sec \theta -\tan \theta =-\frac{a}{b}\left[\because {\sec }^2\theta -{\tan }^2\theta =1\right]\\ \Rightarrow \sec \theta =-\frac{\left(a^2+b^2\right)}{2ab},\tan \theta =\frac{\left(a^2-b^2\right)}{2ab}\\ \Rightarrow \frac{\left(a^2+b^2\right)\left(b^2-a^2\right)}{4a^2{b}^2}=\frac{c}{a}\\ \Rightarrow b^4-a^4=4ac{b}^2\end{array}$